Buddhabrot fractals and non-linear chaotic systemspapañca - diversification; diffuseness; spreading; mental diffusion, mental proliferationComplex non-linear systems:"These systems are apt analogues for dependent co-arising (paticca-samuppada) in that, despite the fact that their behavior sometimes seems chaotic, their behavior actually contains deep, regular patterns. Furthermore, in some cases, these patterns can be utilized so that the system will behave in a desired way." (Thanissaro Bhikkhu)Fractal Definition: A geometric pattern that is repeated (iterated) at ever smaller (or larger) scales to produce (self similar) irregular shapes and surfaces that cannot be represented by classical (Euclidian) geometry. Fractals are used especially in computer modeling of irregular patterns and structures found in nature.(1)...and the amazing story of Mandelbrot Monkhttp://gavesako.blogspot.com/2012/10/bu ... inear.html

Fractal structure of Borobudur and crop circles in EnglandBorobudur Temple was built by Modern Mathematical SciencesAccording Hokky, Borobudur was built space that has a similarity with the elements themselves. In the Borobudur, for example, there are many forms of geometry stupa. “Candi Borobudur stupa itself is a giant in which consists of stupas of other smaller ones. Keep up to infinity,” he explains.http://gavesako.blogspot.com/2012/10/fr ... -crop.html

Fascinating things, fractals. You may be interested to know the Mandelbrot Monk article is a hoax written by a mathematician, for an April Fools joke in a mathematical journal.Putting that story in there uncritically kinda casts a shadow of doubt over some of the more...er... fringe links you have provided in your blog.

Did you ever go to Bodhinyanarama Monastery in New Zealand? Or meet up with a bhikkhu who ended up there? Because they had a most beautiful, detailed book of a recent year's Crop Circle appearances. I thought of you when I remembered that, and wondered if you had given it.

Then, saturated with joy, you will put an end to suffering and stress.SN 9.11

James the Giant wrote:Did you ever go to Bodhinyanarama Monastery in New Zealand? Or meet up with a bhikkhu who ended up there? Because they had a most beautiful, detailed book of a recent year's Crop Circle appearances. I thought of you when I remembered that, and wondered if you had given it.

[/quote]Ven Gavesako did live there. Unfortunately I never met him when he came down here...

It's a great hoax story, the Mandelbrot Monk, and it could almost make another "Da Vinci Code" fiction book...

The hoax was lent an air of credibility because often medieval monks did discover scientific and mathematical theories, only to have them hidden or shelved due to persecution or simply ignored because publication prior to the invention of the printing press was difficult at best. Mr. Girvan adds to this suggestion by associating Udo with several other more legitimate discoveries where an author was considered ahead of his time in terms of a scientific theory of some sort that is now established as a mainstream theory but was considered fringe science at the time.Another aspect of the deception was that it was very common for pre-20th century mathematicians to spend incredible amounts of time on hand calculations such as a logarithm table or trigonometric functions. Calculating all of the points for a Mandelbrot set is a comparable activity that would seem tedious today but would be routine for people of the time.

The funny part is where the monk disagrees with his friend about the nature of the Mandelbrot: one takes it to be a "fingerprint of God's creation" and the other maintains that it is a trick of the Devil which is supposed to take the souls further away from salvation. This actually goes back to some serious theological debates among the Christian scholastics: whether one really possesses free will or everything is pre-determined from the beginning (some of these views being regarded as heresies). Interestingly, the same view is ascribed in the Suttas to one of the philosophers, Makkhali Gosala:

""'Though one might think, "Through this morality, this practice, this austerity, or thisholy life I will ripen unripened kamma and eliminate ripened kammawhenever touched by it" — that is impossible. Pleasure and pain aremeasured out, the wandering-on (samsara) is fixed in its limits. There is noshortening or lengthening, no accelerating or decelerating. Just as aball of string, when thrown, comes to its end simply by unwinding, inthe same way, having transmigrated and wandered on, the wise and thefoolish alike will put an end to pain.'"

I never had a book on crop circles and barely noticed such phenomena, but recently they have been reported in our vicinity and talked about, so I looked them up and some of the geometrical shapes also reminded me of the fractal structures (however they were created). But don't worry, I don't believe in UFOs and alien abductions.

Far from being a mere curiosity, the phenomena of self-similarity and recursiveness have been noticed by Buddhist writers such as Samanera Bodhesako. He wrote in his essay "The Buddha and Catch-22":

And the reason for this, the Buddha informs us, is because of avijjā, or ignorance. But avijjā is not a mere absence of information; it is a refusal to see what is at all times there to be seen. It is not failure to see one particular thing among other particular things, but a radical refusal to see the way all particular things are, and in this respect it is as great a modifier as death — indeed, the two are (so the Buddha tells us) inseparable. The dependent arising formulation says, in summary, “With ignorance as condition, ageing and death, sorrow, lamentation, pain, grief, and despair come into being.”

The deluded person, in refusing to see the nature of all things, refuses also to see the nature of his refusal to see (which is also a thing). That is, he refuses to see delusion. Thus, by denying itself delusion sustains itself. This is stated in the Suttas (e.g. Sammāditthi Sutta, M. 9) as follows:

Friends, that which is non-knowledge of suffering, non-knowledge of the arising of suffering, non-knowledge of the ceasing of suffering, non-knowledge of the way leading to the ceasing of suffering, this, friends, is called ignorance.

For after all, what is “the way leading to the ceasing of suffering”? It is (the Suttas tell us) the noble eightfold path. And what is the first factor of this path? Right view. Ignorance, then, involves non-knowledge of right view. And right view is knowledge of the arising of suffering; that is to say, knowledge of ignorance. Right view is knowledge of right view, and also knowledge of wrong view, whereas wrong view is non-knowledge of wrong view, and also non-knowledge of right view. And this structure of ignorance is, in fact, Catch-22 at its most fundamental level:

Yathābhūtam na pajānāti: he does not see things as they really are: the phrase — so typical a Sutta description of the puthujjana, the unenlightened commoner — is used here by Heller to illuminate precisely the characteristic of being entrapped in a situation. Not only does the puthujjana have flies in his eyes, he does not see that he has them, and he does not see this because he has them. His dilemma is that though he must find a way to see, yet he cannot find that way precisely because he cannot see. Indeed, he cannot even see for himself that this is his problem. And this is the dilemma which, at its most fundamental level, is the specific concern of the Buddha’s Teaching. The structure of avijjā, the structure of Catch-22, the structure of “having flies in one’s eyes”: they are one and the same. Catch-22 is avijjā. The title character in both the novel and in our lives never appears and yet is omnipresent.

He also explored the theme of endless repetition in "The Myth of Sisyphus: A Cycle":

“The work is endless!” exclaimed Anuruddha. “No end to the work is apparent. When does the work conclude? When is an end to the work apparent? When will we be able to indulge ourselves carelessly in the pleasures of the five senses?”

“But, dear Anuruddha, the work is indeed endless. No end to the work is apparent. Even when our fathers and grandfathers died the work did not cease.”

“Well then,” decided Anuruddha, “you know about the duties of the household life. I will go forth from home into homelessness.”

And he wrote a whole book called "Change" which goes into a lot of detail about this topic of recursiveness as found in the Suttas taught by the Buddha. Here is my favourite chapter which approaches the subject with a humorous example:

You and I would have no difficulty in accepting the statement “all circles are round.” It is obvious. Indeed, it is virtually a pleonasm. True, we have not inspected every circle that exists and tested each for roundness. True, we may have personally come across but a minute fraction of all circles that presently exist (let alone those that have been or will come to be). And yet this introduces no jot of doubt into our conviction that all circles are in fact round. Our certainty is structural, not statistical. ...

We saw that experience was hierarchical in its general outlines; we now discover that within experience there exist autonomous hierarchical structures. In the experiential hierarchy “notes — song — concert — evening” the content determined the level within the hierarchy. Notes is more immediate than song and cannot be otherwise. But in the hierarchy of self-deception denial of knowledge is found on every level, and thus describes not a particular level but the hierarchy as a whole. Such hierarchies can be described as replicative, or as recursive.[19]

Recursiveness is important because it offers a stability not present in “ordinary” hierarchies. Remove “an evening on the town” and the entire structure — notes, song, concert — collapses. But remove “denial of knowledge” and we find that…we can’t. Recursiveness is not a feature found merely on each level, like the identical floral pattern on each dish in a stack: rather, recursion is the link between adjacent levels. The denial is always on the next most general level to the knowledge. From the perspective of the knowledge, then, the denial is extra-temporal. As long as we fail to achieve a point of view established outside this hierarchy, knowledge can never escape being encompassed by denial, and the structure must remain inviolable. Thus the structure of self-deception has a stability not found in non-recursive aspects of experience — as everyone knows who has ever succeeded in freeing himself from even the narrowest of such deceptions.

But why go to the trouble of so much self-deception? Why should we be so reluctant to acknowledge the necessity, in experience, of impermanence, when we feel no such hesitation in asserting the necessity, in circles, of roundness? The answer will be found reflected in the entire history of humankind. We seek happiness. We seek freedom. We seek security. Or, more fundamentally, we seek. And so we return, as we must, to craving.

Bodhesako was writing back in the '80s when computers were just starting and he was aware of some philosophical issues surrounding them:

19. The word might be defined, dictionary style, as: “Recursive: adj. see Recursive.” Curiously, recursive hierarchies seem to play an important role in some branches of Western science, including computer programming, wherein it is essential that such programs do not contain any true recursive hierarchies. For if even one were to be introduced the computer would become involved in an endless cycle and the program would never conclude. In other words, although art may imitate life, a computer program, if it is ever to arrive at a conclusion, had better not do so too closely.The term “recursive” (as well as several other words) has been adapted with a somewhat altered meaning from Douglas R. Hofstadter’s Gödel, Escher, Bach: An Eternal Golden Braid (Penguin Books, 1980). Hofstadter’s book is provocative, witty, imaginative, wide-ranging, entertaining, stimulating, and, alas, quite mistaken in its fundamental approach to understanding the human situation. Neither his deterministic views nor (at the other extreme) the free-will views of Prof. J. R. Lucas can come close to the middle way taught by the Buddha.

I think it would be hard to find a better visual demonstration of recursiveness than this (ignore the mathematical formulas at the end if you cannot grasp the abstract concepts -- seeing it is enough):Mandelbrot sethttp://www.youtube.com/watch?v=UuOTuFVnWv0

Complex non-linear systems:"These systems are apt analogues for dependent co-arising (paticca-samuppada) in that, despite the fact that their behavior sometimes seems chaotic, their behavior actually contains deep, regular patterns. Furthermore, in some cases, these patterns can be utilized so that the system will behave in a desired way." (Thanissaro Bhikkhu)http://www.scribd.com/doc/103907285/Shape-of-Suffering

The dictionary definition of chaos is turmoil, turbulence, primordial abyss, and undesired randomness, but scientists will tell you that chaos is something extremely sensitive to initial conditions. Chaos also refers to the question of whether or not it is possible to make good long-term predictions about how a system will act. A chaotic system can actually develop in a way that appears very smooth and ordered. (Chaos Theory and Fractals)http://www.tnellen.com/alt/chaos.html

Thanissaro Bhikkhu also talks about self-similarity and scale invariance. E.g.

The Shape of Suffering: A Study of Dependent Co-arising, by Thanissaro Bhikkhu In this book the author examines the nature of dependent co-arising (paticca samuppada) — the complex causal structure by which dukkha arises and ceases. It also shows how the factors of the path address the causes of suffering in a way that leads to its cessation. http://www.accesstoinsight.org/lib/auth ... index.html

I have made similar use of modern science — chaos theory in particular. There are many parallels between Buddhist theories of causation and modern deterministic chaos theory. Examples and terminology drawn from the latter — such as feedback, scale invariance, resonance, and fluid turbulence — are very useful in explaining the former. Again, in using these parallels I am not trying to equate Buddhist teachings with chaos theory or to engage in pseudo-science. Fashions in science change so rapidly that we do the Buddha's teachings no favor in trying to "prove" them in light of current scientific paradigms. Here I am simply pointing out similarities as a way of helping to make those teachings intelligible in modern terms. Deterministic chaos theory is the only modern body of knowledge that has worked out a vocabulary for the patterns of behavior described in Buddhist explanations of causality, and so it seems a natural source to draw on, both to describe those patterns and to point out some of their less obvious implications.

The non-linearity of this/that conditionality explains why heightened skillfulness, when focused on the present moment, can succeed in leading to the end of the kamma that has formed the experience of the entire cosmos. All non-linear processes exhibit what is called scale invariance, which means that the behavior of the process on any one scale is similar to its behavior on smaller or larger scales. To understand, say, the large-scale pattern of a particular non-linear process, one need only focus on its behavior on a smaller scale that is easier to observe, and one will see the same pattern at work. In the case of kamma, one need only focus on the process of kamma in the immediate present, in the course of developing heightened skillfulness, and the large-scale issues over the expanses of space and time will become clear as one gains release from them.

The teaching on dependent co-arising helps to provide more detailed instructions on this point, showing precisely where the cycle of kamma provides openings for more skillful present input. In doing so, it both explains the importance of the act of attention in developing the fourth type of kamma, and acts as a guide for focusing attention on present experience in appropriate ways [III/H/iii].

Dependent co-arising shows how the cosmos, when viewed in the context of how it is directly experienced by a person developing skillfulness, is subsumed entirely under factors that are immediately present to awareness: the five aggregates of form, feeling, perception, mental fabrication, and consciousness, and the six sense media [§§212-213]. Included in this description is the Buddha's ultimate analysis of kamma and rebirth. The nexus of kamma, clinging, becoming, and birth accounts for the realm in which birth takes place [§220], whereas the nexus of name-and-form with consciousness accounts for the arising and survival of the kammically active organism within that realm [§231]. Also included in dependent co-arising is a detailed analysis of the way in which kamma can — but does not necessarily have to — lead to bondage to the cycle of rebirth. Unlike the Jains, the Buddha taught that this bondage was mental rather than physical. It was caused not by sticky substances created by the physical violence of an act, but by the fact that, when there is ignorance of the four noble truths [III/H/i] (a subtle form of delusion, the most basic root of unskillfulness), the feeling that results from kamma gives rise to craving (a subtle form of greed and aversion), clinging, and becoming; and these, in turn, form the conditions for further kamma.

I too find the way in which Thanissaro Bhikkhu evokes the terminology of complexity/systems theory as a kind of poetic language to explicate Buddhist ideas insightful. Early this year I read a book called A World of Becoming by William E. Connolly, political philosopher at John Hopkins University. I'm very taken by it because the notion of becoming threading through the book resonates with some aspects of Buddhist understandings. Connolly engages with various fields of knowledge, from philosophy to biology to neurosciences to systems theory, to argue that the challenges of the contemporary world call for greater attention to a 'micropolitics of becoming'—i.e. how creative adjustments in our everyday modes of perception, bodily habits and ethical sensibilities might generate feedback and resonances that would impact upon broader social, economic, and political systems, which he describes as 'force fields' rather than 'structures'. It seems to me that Connolly is making a similar suggestion as Thanissaro Bhikkhu: 'All non-linear processes exhibit what is called scale invariance, which means that the behavior of the process on any one scale is similar to its behavior on smaller or larger scales. To understand, say, the large-scale pattern of a particular non-linear process, one need only focus on its behavior on a smaller scale that is easier to observe, and one will see the same pattern at work.'

The 'postlude' to the book is interesting, you can read it here on Google books: "Postlude" to A World of Becoming Below is an excerpt from pages 55-6 of the book where he talks about the micropolitics of perception. To my reading, it seems that what we are doing when we engage in meditative exercises—and bhavana in general—is exploring ways to open, dissolve, and steer what Connolly calls 'anticipatory habits and sedimented dispositions'—not unsuitable terms for explicating kamma, I think.

Of course I am partial the multiple-time-scale description of dependent origination, and the way Thanissaro Bhikkhu argues that this is the key to being able to do learn something about it - that we can are able to observe on short time-scales and extrapolate to longer timescales.

I have made similar use of modern science — chaos theory in particular. There are many parallels between Buddhist theories of causation and modern deterministic chaos theory. Examples and terminology drawn from the latter — such as feedback, scale invariance, resonance, and fluid turbulence — are very useful in explaining the former. Again, in using these parallels I am not trying to equate Buddhist teachings with chaos theory or to engage in pseudo-science. Fashions in science change so rapidly that we do the Buddha's teachings no favor in trying to "prove" them in light of current scientific paradigms. Here I am simply pointing out similarities as a way of helping to make those teachings intelligible in modern terms. Deterministic chaos theory is the only modern body of knowledge that has worked out a vocabulary for the patterns of behavior described in Buddhist explanations of causality, and so it seems a natural source to draw on, both to describe those patterns and to point out some of their less obvious implications.

It seems that the Buddha was deliberately applying the principle of non-linear chaotic systems in some of his similes describing the uncertainty of rebirth and of kamma-vipaka in general, because there are so many variables (the initial conditions = input of actions).

Samsara means 'aimless wandering' of one’s life. From lifetime to lifetime we wander in this form or another, or even from day to day, moment to moment. In many Buddhist texts, the Buddha describes this aimless wandering. In the Danda Sutta (SN 15.9), the Buddha says:

From an inconstruable beginning comes transmigration. A beginning point is not evident, though beings hindered by ignorance and fettered by craving are transmigrating and wandering on. Just as a stick thrown up in the air lands sometimes on its base, sometimes on its side, sometimes on its tip; in the same way, beings hindered by ignorance and fettered by craving, transmigrating and wandering on, sometimes go from this world to another world, sometimes come from another world to this.

The simile of the stick is important. We assume that good kamma = good vipaka (results), or even a better rebirth, but the issue of kamma is insanely complicated. In another, much longer sutra, the Mahakammavibhanga Sutta (MN 136) or “The Greater Exposition on Kamma”, the Buddha explains that even if you do good deeds now, you can still have a bad rebirth because you also have to take into account all the kamma you’ve committed in past lives. The good kamma you commit will come to fruition someday, but not necessarily when you think it will. For something like rebirth, many actions, thoughts and such all take into account and come to fruition as one dies and is reborn again.

“It is kamma which makes the world go round; Kamma keeps moving on the lives of men. All beings to kamma are bound, as a linchpin to the chariot wheel.” (Sutta Nipata)

It could also be compared to double pendulum which is a wonderful example of chaos theory, how something that seems so simple can be so unpredictable. A double pendulum is pretty much what it sounds like: you attach one pendulum to the end of another, making it a double-jointed pendulum. But unlike a normal pendulum, the double pendulum is sensitive to initial conditions and never does the same thing twice: http://pippagoldenberg.wordpress.com/20 ... -pendulum/

Another example involves the turbulences in a flow of water which is non-linear chaotic process in nature:

"Imagine that the whole earth was covered with water, and that a man were to throw a yoke with a hole in it into the water. Blown by the wind, that yoke would drift north, south, east and west. Now suppose that once in a hundred years a blind turtle were to rise to the surface. What would be the chances of that turtle putting his head through the hole in the yoke as he rose to the surface once in a hundred years?""It would be very unlikely, Lord.""Well, it is just as unlikely that one will be born as a human being. It is just as unlikely that a Tathagata, a Noble One, a fully enlightened Buddha should appear in the world. And it is just as unlikely that the Dhamma and discipline of the Tathagata should be proclaimed. But now you have been born as a human being, a Tathagata has appeared and the Dhamma has been proclaimed. Therefore, strive to realize the Four Noble Truths." (S.V,456)

This interesting passage uses the metaphor of a dice game -- with its incalculable probability -- to illustrate the workings of kamma. What the simile seems to imply is that, despite the unpredictability, the results of good and bad actions will always in the end bring corresponding fruition.

"As to that tainted failure (in living), which is threefold in bodily acts, fourfold in verbal acts and threefold in mental acts, and which, having been caused by unwholesome volition, produces suffering, results in suffering, it is due to those very failures (in living) that beings, after death, when the body breaks up, are reborn in a world of woe, in an unhappy destiny, a state of misery, in the hells."Just as a perfect throw of dice, when thrown upwards, will come to rest firmly wherever it falls, similarly beings will be reborn in states of woe, in an unhappy destiny, a state of misery, in the hells, due to those tainted failures (in living) caused by unwholesome volition."I declare, monks, that actions (kamma) willed, performed and accumulated, will not become extinct as long as their results have not been experienced, be it in this life, in the next life or in future lives. And as long as these results of actions willed, performed and accumulated, have not been experienced, there will be no end to suffering, I declare. ..."As to that success (in living), which is threefold in bodily acts, fourfold in verbal acts and threefold in mental acts, and having been caused by wholesome volition, produces happiness, results in happiness, it is due to that very success (in living) that beings after death, on the break-up of the body, are reborn to a happy destiny, in a heavenly world."Just as a perfect throw of dice, when thrown upwards, will come to rest firmly wherever it falls, similarly beings will be reborn to a happy destiny, in a heavenly world, due to success (in living), caused by wholesome volition."I declare, monks, that actions (kamma) willed, performed and accumulated, will not become extinct as long as their results have not been experienced, be it in this life, in the next life or in future lives. And as long as these results of actions willed, performed and accumulated, have not been experienced, there will be no end to suffering, I declare." (AN 10.206)

This video was made to illustrate the Buddha’s teaching on dependent co-arising or dependent origination (paticca-samuppāda) which can be observed at many scales, which means that lessons drawn from observing the world (loka) can be applied to our internal experience of the six senses which makes up ‘our world’. The black hole at the center of a galaxy is like the fundamental ignorance which gives rise to the formations that construct our experience in terms of consciousness being established in name & form. The Buddha compares the ongoing cycle of becoming to the process of planting seeds in fertile soil which is watered by rain.

Compare this recent report about the large-scale chaotic systems and their behaviour, similar to the small-scale:

Graceful in their turnings, spiral galaxies were thought to have reached their current state billions of years ago. A study of hundreds of galaxies, however, upsets that notion revealing that spiral galaxies, like the Andromeda Galaxy and our own Milky Way, have continued to change.Farther back in time, galaxies tend to be very different, say astronomers, with random and disorganized motions. Nearer to the present, star-forming galaxies look like well-ordered disk-shaped systems. Rotation in these galaxies trumps other internal, random motions. These galaxies are gradually settling into well-behaved disks with the most massive galaxies always showing higher organization.The Milky Way Galaxy may have gone through the same chaotic growing and changing as the galaxies in the DEEP2 sample before settling into its present state at just about the same time the Sun and Earth were forming, say team scientists. By observing the pattern, astronomers can now adjust computer simulations of galaxy evolution until they replicate the observations. Then the hunt will be on to determine the physical processes responsible for the trend.